Mounts have always suffered from the way DnD is set up, and are actually impractical in higher tier of play.
So we have redesign their health pool to accompany and scale with the rider.
The mount will now have the same amount of hitdie as your character, let's demonstrate with the Warhorse.
The Warhorse is a large beast, with 11 AC and 19(3d10 + 3), so that would be its default health pool for a character of third level.
For higher levels?
Just add an extra d10 and take the average (since it's what we use for the characters health calculation).
In the case of a d10 the average would be 6 + constitution modifier (3), thus a level four character's mount would have +9 to their health pool, summing up to 28 hitpoints, rinse and repeat this as your character levels.
How to apply this to initiative?
Once initiative has been declared by your resident Dungeon Master, after you have joined initiative as your character, add your mount by typing the following once initiative has begun. (Make sure you are aware of the amount of health your mount currently has at your level)
!i madd warhorse -hp 28 -p value
On the value area type your CHARACTERS initiative roll. (so you may take your turns in unison with your main character)
Additionally should you buy barding (mount armour) you can also add it by typing the following. (For the purpose of this demonstration I'm going to say I bought plate armour for my mount (AC 18) coating four times the amount needed fora regular plate if armour, summing up to 6,000gp)
!i madd warhorse -hp 28 -ac 28 -p value
To attack as your mount it's fairly simple
(You can check its attack options by typing !i a list on the mounts turn, or have its stats in hand from a website such as 5e.tools)
!i a hooves -t targetname
Happy riding! And name your mount! It is likely the two of you will embark on awesome adventurers!
Kilroy